Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Abstract

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,…,tk. A formula in this language defines a parametric set St⊆Zd as t varies in Zk, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming P≠NP) we construct a parametric set St1,t2 such that |St1,t2| is not even polynomial‐time computable on input (t1,t2). In contrast, for parametric sets St⊆Zd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that |St| is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.